Question

If ${{C}_{1}},{{C}_{2}},{{C}_{3}},.....$represent the speeds of ${{n}_{1}},{{n}_{2}},{{n}_{3}},.....$molecules, then the root mean square speed is [IIT$1993$]
A.${{\left( \dfrac{{{n}_{1}}C_{1}^{2}+{{n}_{2}}C_{2}^{2}+{{n}_{3}}C_{3}^{2}+.....}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....} \right)}^{\dfrac{1}{2}}}$
B.$\dfrac{{{({{n}_{1}}C_{1}^{2}+{{n}_{2}}C_{2}^{2}+{{n}_{3}}C_{3}^{2}+.....)}^{\dfrac{1}{2}}}}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}$
C.$\dfrac{{{\left( {{n}_{1}}C_{1}^{2} \right)}^{\dfrac{1}{2}}}}{{{n}_{1}}}+\dfrac{{{\left( {{n}_{2}}C_{2}^{2} \right)}^{\dfrac{1}{2}}}}{{{n}_{2}}}+\dfrac{{{\left( {{n}_{3}}C_{3}^{2} \right)}^{\dfrac{1}{2}}}}{{{n}_{3}}}+.....$
D.${{\left[ \dfrac{{{({{n}_{1}}C_{1}^{2}+{{n}_{2}}C_{2}^{2}+{{n}_{3}}C_{3}^{2}+.....)}^{2}}}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....} \right]}^{\dfrac{1}{2}}}$

Answer 1

Hint: The root mean square speed is the square root of the mean of the squares of the different velocities of the gaseous molecules. So, at first, we will have to calculate the mean of the squares of different velocities by using the given data.

Complete step by step solution:According to the kinetic theory of gas, gasses consist of different atoms or molecules that randomly move at different speeds in any direction. Therefore these gaseous particles possess different speeds. Here we have to calculate the root mean square speed of gaseous molecules. This can be calculated by the sum of squares of the individual speed values divided by the total number of gas molecules present.
Their ${{n}_{1}},{{n}_{2}},{{n}_{3}},.....$molecules possess speed ${{C}_{1}},{{C}_{2}},{{C}_{3}},.....$ which means ${{n}_{1}}$molecules have speed ${{C}_{1}}$, ${{n}_{2}}$molecules have speed ${{C}_{2}}$,${{n}_{3}}$molecules have speed ${{C}_{3}}$, and so on.
Hence the total sum of squares of the speeds of each molecule is:
$({{n}_{1}}C_{1}^{2}+{{n}_{2}}C_{2}^{2}+{{n}_{3}}C_{3}^{2}+.....)$
And the total number of molecules $=({{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....)$
$\therefore $The mean of the squares of different speeds $=\dfrac{{{n}_{1}}C_{1}^{2}+{{n}_{2}}C_{2}^{2}+{{n}_{3}}C_{3}^{2}+.....}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}$
Therefore square root of the mean of the squares of speeds is:
$\sqrt{\dfrac{{{n}_{1}}C_{1}^{2}+{{n}_{2}}C_{2}^{2}+{{n}_{3}}C_{3}^{2}+.....}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....}}$ or,${{\left[ \dfrac{{{n}_{1}}C_{1}^{2}+{{n}_{2}}C_{2}^{2}+{{n}_{3}}C_{3}^{2}+.....}{{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....} \right]}^{\dfrac{1}{2}}}$

Thus, option (A) is correct.
Additional Information:The Maxwell-Boltzmann equation defines the speed distribution for gas at a particular temperature. The most likely speed, average velocity, root mean square velocity and most probable velocity all can be calculated using this distribution function.
According to the Maxwell-Boltzman distribution equation, average velocity, root mean square velocity and most probable velocity can be calculated when the temperature$(T)$and molecular mass$(M)$of gaseous molecules are known.
Average velocity$=\sqrt{\dfrac{8RT}{\pi M}}$
Root mean square velocity$=\sqrt{\dfrac{3RT}{M}}$
Most probable velocity$=\sqrt{\dfrac{2RT}{M}}$

Note: According to the Maxwell-Boltzmann distribution equation, the ratio of average velocity, root mean square velocity, and most probable velocity is $1.128:1.224:1$. Root mean square velocity is always greater than the average velocity and most probable velocity of gas molecules.